> c i so that delay until at least t + 1 is preferred. To verify when t + 1 is less than t compute: tf-^-n + rfif^-i) ( - ^ -V r \pw J \v — k — (1 4- r)n/r clog(A<5) J As before, continuity of the optimal time in c means there is a c2 such that -6 \og(X)(v - k - (1 + r)7r/r) < c 2 < - Iog(A)0 - k - (1 + r)-n/r) and for 4> = c2, the good type is indifferent to investing at the closest date before t and the closest date after i. Thus for 4> > c2, the good type's optimal investment time is i and then R* = v. Existence The next part of the proof shows that there exists an equilibrium in which the good type invests at the t* determined above with an offer price of i r . This is done by construction. Consider the following equilibrium {s*,(j,*}: . 0 if t < t* Se(t,G) Rl if t > t* se(t, 5) = 0 for all t H*(G\t,R) sm(t,R) = { 1 if t > t* and R = Rt 0 otherwise. 1 if (t > t* and R = R') or R < pw 0 otherwise. Given the market's strategy, any deviation by the good type before t* will lead to a rejection or selling shares at a price below pw, which in either case gives a lower payoff than the equilibrium payoff. Consider a deviation at any date on or after t*. The good type could raise the offer, which will then be rejected. But at any t>t*, the entrepreneur prefers 46 offering R} and investing than waiting until t+ to invest. Alternatively, pw or lower could be offered, which will be accepted. But Rl > pw. The last possibility is to make no offer and wait until the next date to resume with the equilibrium strategy. But for t > t*, given the derivative (1.23), the good type prefers investing at t with price Rl than waiting until t + 1 to invest at price Rt+l. For the bad type, any deviation that does not mimic the good type's equilibrium offer and is above pw is rejected, which gives the low type of strictly lower payoff (pays /) than not making an offer. Making the same offer as the good type leaves the bad type's payoff unchanged because Rl is determined by the strictly binding IC constraint. Finally, any offer at or below pw cannot be profitable by definition of a bad project. Thus condition 1 of the definition is satisfied for both types. And since the good type makes an offer and the bad type doesn't, condition 3 is satisfied. Since the market believes that if any offer other than Rl at any t > t* is made, the project is bad, it correctly rejects all such offers unless the price is below pw. Accepting the offer R} at any t > t* leaves the market with a non-negative payoff, given its beliefs, and so condition 2 of the definition holds. Last, consider beliefs. Let s be an alternative set of strategies such that at some t the good type offers R < R* and t ^ t*, R^ R}* or both. Since the sign of the derivative (1.23) does not depend on t', we have for any f 0. Points 2. and 3. together show that when a firm underprices it also sells no secondary shares, 7# = 1. However, from proposition 2.2, point 3, we know that when a firm doesn't underprice, some secondary shares are sold. This is an important testable implication of the model: Secondary shares are sold if, and only if, the firm doesn't underprice. We have already seen another testable implication in the example: an increase in K from $5 million to $7 million led to underpricing. Moreover, a further increase in K would have led to even greater underpricing. The following proposition generalizes this result. Proposition 2.3 dRH/dK = 0 ifK